This unit primarily concerns exponential functions and their inverses -- logarithmic functions. Within exponential functions, we will start by covering exponential growth and decay, using real-life scenarios to motivate our discussions. So far we have worked on fivetasks:Steps and Folds, BankingOnYour Success, Steps and Folds Part II, Wrecked!, and The Big Cheese. Through these tasks, students will successfully be able to mathematically describe discrete and continuous exponential growth and decay using multiple representations -- including tabular, algebraic, geometric, and verbal.
In accordance with the Common Core State Standards for Mathematics, lessons are designed so that students consistently address the Standards for Mathematical Practice
Common Core Standards Addressed: CCSS.MATH.CONTENT.HSF.LE.A.1.C - Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. CCSS.MATH.CONTENT.HSF.LE.A.4 - For exponential models, express as a logarithm the solution to ab^(ct)=d where a, c, and d are numbers and b is the base 2, 10, or e.
Unit 5, Lesson 1:
An Introduction to Exponential Functions Task: Steps and Folds Description: In this lesson, we use a fun activity to look at a situation modeled by a linear function and another situation modeled by an exponential function. Using a table, equation, and graph, we compare and contrast the features of the two, such as their rates of change and the role of the independent variable (x in most cases). Homework: Finish U5L1 handout using the information gathered in class. Ensure your separate notesheet with a table and graph for each of the experiments is complete and stored in your binder. Objectives:
C - SWBAT explain differences between linear and exponential functions.
L - SWBAT read a problem, write formulas and facts, and communicate results.
M - SW demonstrate mastery of the objectives by investigating a real life scenario modeled by linear and exponential functions. Lesson Handout Period 5 Presentation Period 6 Presentation
Unit 5, Lesson 2:
Exponential Functions II - Exponential Growth (Discrete) Task: Banking on Your Success Description: After looking at a basic example of an exponential growth function, we look at a more complicated example. Using a scenario of a savings account that is opened to save up for "something nice" (e.x. Playstation 4/iPhone 6), students work together to search for a pattern that will easily allow us to determine the value that the bank account accrues after any number of months. Students are introduced to the general formula for exponential growth given a rate r, an initial amount a, and a period of time n (or sometimes t). Homework: Finish U5L2 handout using what we discovered during class. Complete the U5L2 homework: Revving Up for Next Time!. For this homework, look up the price of a car you really like online. Provide the make and model of the car, the new price of the car, and the source the information came from. Objectives:
C - SWBAT apply the concept of exponential growth to a real-life scenario.
M - SW demonstrate mastery of the objective by investigating a real-life situation using the principles of exponential growth. Lesson Handout Period 5 Presentation Period 6 Presentation
Unit 5, Lesson 3:
Exponential Functions III - Exponential Decay (Discrete) Tasks: Cuts and Folds, Part II & Wrecked! Description: After discovering an easy way to quickly calculate exponential growth through Bankingon Your Success, students use that knowledge to investigate exponential decay. Going back to Cuts and Folds, we watch an episode of Mythbusters in which they investigate whether a piece of paper can be folded in half more than eight times. This leads into a discussion with students about how the number of folds is related to the number of rectangles the paper is divided into or the fraction of the entire area each of the rectangles accounts for. Students then use the information they gathered from U5L2 homework to do Wrecked, where they model car depreciation using an exponential function. Through these activities, students discover the formula for discrete exponential decay. Objectives:
C - SWBAT apply the concept of exponential decay to a real-life scenario.
M - SW demonstrate mastery of the objective by investigating a real-life situation using the principles of exponential decay. Lesson Handout (Page 1) Lesson Handout (Page 2) Period 5 Presentation(DAY 1) Period 5 Presentation(DAY 2) Period 6 Presentation(DAY 1) Period 6 Presentation(DAY 2)
Unit 5, Lesson 4:
Exponential Functions IV - Continuous Exponential Growth Task: The Big Cheese Description: Now that students have practiced discrete exponential growth and decay through Cuts and Folds,Bankingon Your Success, Cuts and Folds Part II and Wrecked!, students will examine the effect of calculating exponential growth over various intervals. In The Big Cheese, students will investigate the impact on calculating a yearly raise based on a yearly increase, a monthly increase, a daily increase, and an hourly increase. In doing so, they will examine patterns involving the interest rate being applied, in addition to how this impacts the outcome of the situation. They will also need to write functions for each type of exponential growth. This lesson leads to the discovery of e, which will then pave the way for an investigation into continuous decay and logarithmic functions. Objectives:
C - SWBAT calculate discrete and continuous exponential growth using functions that model a real-life scenario.
M - SW demonstrate mastery of the objective by investigating a real-life situation. Lesson Handout Periods 5 and 6 Presentation
Unit 5, Lesson 5:
Logarithmic Functions Task: An Ancient Puzzle Description: Previously, we have studied exponential functions, and how they can grow or decay. Now, we begin an investigation of the inverses of exponential functions. With exponential functions, we are able to obtain output values when given a base raised to a power x. Logarithmic functions answer the question: given a base (such as 2 or 10), and a value y, what power x do we need to raise the base to in order to get y. In this lesson, we begin to analyze these functions through geometric, algebraic, tabular, and verbal approaches. Objectives:
C - SWBAT convert between exponential and logarithmic functions by hand and using technology.
M - SW demonstrate mastery of the objective by analyzing exponential and logarithmic functions using a table, graph, function, and verbal description. Lesson Handout Period 5 and 6 Presentation
Unit 5, Quiz 1 Review:
Review for Quiz 1 Task: How HOT Are You? Description: In preparation for the pre-break assessment, we will be doing a stations review activity. Students will be responsible for creating their own study guide, which will factor into their assessment grade. They will create the study guide by assessing their own knowledge of the U5L1-U5L4 content, then committing to a plan to best prepare themselves for the assessment. Objectives:
C - SWBAT review for a quiz on discrete and continuous exponential growth and decay.
M - SW demonstrate mastery of the objective by creating a study plan to be used to prepare for U5Q1. Lesson Handout
Unit 5 - Exponential Functions and Their Inverses
This unit primarily concerns exponential functions and their inverses -- logarithmic functions. Within exponential functions, we will start by covering exponential growth and decay, using real-life scenarios to motivate our discussions. So far we have worked on five tasks: Steps and Folds, Banking On Your Success, Steps and Folds Part II, Wrecked!, and The Big Cheese. Through these tasks, students will successfully be able to mathematically describe discrete and continuous exponential growth and decay using multiple representations -- including tabular, algebraic, geometric, and verbal.
In accordance with the Common Core State Standards for Mathematics, lessons are designed so that students consistently address the Standards for Mathematical Practice
Common Core Standards Addressed:
CCSS.MATH.CONTENT.HSF.LE.A.1.C - Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
CCSS.MATH.CONTENT.HSF.LE.A.4 - For exponential models, express as a logarithm the solution to ab^(ct)=d where a, c, and d are numbers and b is the base 2, 10, or e.
Unit 5, Lesson 1:
An Introduction to Exponential FunctionsTask: Steps and Folds
Description: In this lesson, we use a fun activity to look at a situation modeled by a linear function and another situation modeled by an exponential function. Using a table, equation, and graph, we compare and contrast the features of the two, such as their rates of change and the role of the independent variable (x in most cases).
Homework: Finish U5L1 handout using the information gathered in class. Ensure your separate notesheet with a table and graph for each of the experiments is complete and stored in your binder.
Objectives:
C - SWBAT explain differences between linear and exponential functions.
L - SWBAT read a problem, write formulas and facts, and communicate results.
M - SW demonstrate mastery of the objectives by investigating a real life scenario modeled by linear and exponential functions.
Lesson Handout
Period 5 Presentation
Period 6 Presentation
Unit 5, Lesson 2:
Exponential Functions II - Exponential Growth (Discrete)Task: Banking on Your Success
Description: After looking at a basic example of an exponential growth function, we look at a more complicated example. Using a scenario of a savings account that is opened to save up for "something nice" (e.x. Playstation 4/iPhone 6), students work together to search for a pattern that will easily allow us to determine the value that the bank account accrues after any number of months. Students are introduced to the general formula for exponential growth given a rate r, an initial amount a, and a period of time n (or sometimes t).
Homework: Finish U5L2 handout using what we discovered during class. Complete the U5L2 homework: Revving Up for Next Time!. For this homework, look up the price of a car you really like online. Provide the make and model of the car, the new price of the car, and the source the information came from.
Objectives:
C - SWBAT apply the concept of exponential growth to a real-life scenario.
M - SW demonstrate mastery of the objective by investigating a real-life situation using the principles of exponential growth.
Lesson Handout
Period 5 Presentation
Period 6 Presentation
Unit 5, Lesson 3:
Exponential Functions III - Exponential Decay (Discrete)Tasks: Cuts and Folds, Part II & Wrecked!
Description: After discovering an easy way to quickly calculate exponential growth through Banking on Your Success, students use that knowledge to investigate exponential decay. Going back to Cuts and Folds, we watch an episode of Mythbusters in which they investigate whether a piece of paper can be folded in half more than eight times. This leads into a discussion with students about how the number of folds is related to the number of rectangles the paper is divided into or the fraction of the entire area each of the rectangles accounts for. Students then use the information they gathered from U5L2 homework to do Wrecked, where they model car depreciation using an exponential function. Through these activities, students discover the formula for discrete exponential decay.
Objectives:
C - SWBAT apply the concept of exponential decay to a real-life scenario.
M - SW demonstrate mastery of the objective by investigating a real-life situation using the principles of exponential decay.
Lesson Handout (Page 1)
Lesson Handout (Page 2)
Period 5 Presentation (DAY 1)
Period 5 Presentation (DAY 2)
Period 6 Presentation (DAY 1)
Period 6 Presentation (DAY 2)
Unit 5, Lesson 4:
Exponential Functions IV - Continuous Exponential GrowthTask: The Big Cheese
Description: Now that students have practiced discrete exponential growth and decay through Cuts and Folds, Banking on Your Success, Cuts and Folds Part II and Wrecked!, students will examine the effect of calculating exponential growth over various intervals. In The Big Cheese, students will investigate the impact on calculating a yearly raise based on a yearly increase, a monthly increase, a daily increase, and an hourly increase. In doing so, they will examine patterns involving the interest rate being applied, in addition to how this impacts the outcome of the situation. They will also need to write functions for each type of exponential growth. This lesson leads to the discovery of e, which will then pave the way for an investigation into continuous decay and logarithmic functions.
Objectives:
C - SWBAT calculate discrete and continuous exponential growth using functions that model a real-life scenario.
M - SW demonstrate mastery of the objective by investigating a real-life situation.
Lesson Handout
Periods 5 and 6 Presentation
Unit 5, Lesson 5:
Logarithmic FunctionsTask: An Ancient Puzzle
Description: Previously, we have studied exponential functions, and how they can grow or decay. Now, we begin an investigation of the inverses of exponential functions. With exponential functions, we are able to obtain output values when given a base raised to a power x. Logarithmic functions answer the question: given a base (such as 2 or 10), and a value y, what power x do we need to raise the base to in order to get y. In this lesson, we begin to analyze these functions through geometric, algebraic, tabular, and verbal approaches.
Objectives:
C - SWBAT convert between exponential and logarithmic functions by hand and using technology.
M - SW demonstrate mastery of the objective by analyzing exponential and logarithmic functions using a table, graph, function, and verbal description.
Lesson Handout
Period 5 and 6 Presentation
Unit 5, Quiz 1 Review:
Review for Quiz 1Task: How HOT Are You?
Description: In preparation for the pre-break assessment, we will be doing a stations review activity. Students will be responsible for creating their own study guide, which will factor into their assessment grade. They will create the study guide by assessing their own knowledge of the U5L1-U5L4 content, then committing to a plan to best prepare themselves for the assessment.
Objectives:
C - SWBAT review for a quiz on discrete and continuous exponential growth and decay.
M - SW demonstrate mastery of the objective by creating a study plan to be used to prepare for U5Q1.
Lesson Handout
For All Assignments after this go to Mr. Lippman's Assignments Page.